This chapter discusses three kinds of reasoning that illustrate the central role of causality. The first part shows that causal considerations enter into how people think about some mathematical ...
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This chapter discusses three kinds of reasoning that illustrate the central role of causality. The first part shows that causal considerations enter into how people think about some mathematical equations, that their thinking about equations reflects an underlying causal structure. The second part concerns social attribution, reasoning about why people do things. It is shown that people's explanations of behaviour depend on the causal model that they believe governs that behaviour. The third part concerns reasoning about counterfactual events.Less

Reasoning About Causation

Steven Sloman

Published in print: 2005-08-18

This chapter discusses three kinds of reasoning that illustrate the central role of causality. The first part shows that causal considerations enter into how people think about some mathematical equations, that their thinking about equations reflects an underlying causal structure. The second part concerns social attribution, reasoning about why people do things. It is shown that people's explanations of behaviour depend on the causal model that they believe governs that behaviour. The third part concerns reasoning about counterfactual events.

Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To ...
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Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To give a mathematical description of a real phenomenon requires bridge principles. However, given the constraints of theory, even these employ highly idealized fictional objects and processes, more akin to artful theatrical distortions than to true descriptions of things in the world.Less

Fitting Facts to Equations

Nancy Cartwright

Published in print: 1983-06-09

Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To give a mathematical description of a real phenomenon requires bridge principles. However, given the constraints of theory, even these employ highly idealized fictional objects and processes, more akin to artful theatrical distortions than to true descriptions of things in the world.

Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical ...
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Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics—a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.Less

Mathematical Foundations of Quantum Mechanics : New Edition

Published in print: 2018-02-27

Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics—a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.

This chapter continues the discussion began in Chapter α‎. It presents the second part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the ...
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This chapter continues the discussion began in Chapter α‎. It presents the second part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with equations of degree 3 (or 4) in one variable or degree 2 in two variables. He says that if we are willing to allow square roots into our arithmetic, we can consider the quadratic equation a problem whose solution has been long understood (in some cases by the ancient Babylonians). Equations of degree 3 and 4, such as x3 − 2x2 + 14x + 9 and x4 + 5x3 + 11x2 + 17x − 29, were first solved in Renaissance Italy to great acclaim; the solutions are given by formulas involving cube roots and fourth roots.Less

How to Explain Number Theory at a Dinner Party

Michael Harris

Published in print: 2017-05-30

This chapter continues the discussion began in Chapter α‎. It presents the second part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with equations of degree 3 (or 4) in one variable or degree 2 in two variables. He says that if we are willing to allow square roots into our arithmetic, we can consider the quadratic equation a problem whose solution has been long understood (in some cases by the ancient Babylonians). Equations of degree 3 and 4, such as x3 − 2x2 + 14x + 9 and x4 + 5x3 + 11x2 + 17x − 29, were first solved in Renaissance Italy to great acclaim; the solutions are given by formulas involving cube roots and fourth roots.

This chapter illustrates the computations and variations of the electron density distribution of unit cells. To enable computations, crystallographers use electron density maps which are figured as a ...
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This chapter illustrates the computations and variations of the electron density distribution of unit cells. To enable computations, crystallographers use electron density maps which are figured as a mathematical equation of square average value and standard deviation within a single unit cell. Chemical interpretations are derived from these maps and interpretability is dependent on crystallographic resolution. the chapter also examines the analyses on the effectivity of molecular models with the measured diffraction data. Sample problems are provided at the end of the chapter.Less

Electron Density Maps and Molecular Structures

Peter B. Moore

Published in print: 2012-04-19

This chapter illustrates the computations and variations of the electron density distribution of unit cells. To enable computations, crystallographers use electron density maps which are figured as a mathematical equation of square average value and standard deviation within a single unit cell. Chemical interpretations are derived from these maps and interpretability is dependent on crystallographic resolution. the chapter also examines the analyses on the effectivity of molecular models with the measured diffraction data. Sample problems are provided at the end of the chapter.